If `T_(n)=cos^(n)theta+sin ^(n)theta, then 2T_(6)-3T_(4)+1=`

Options

A. 2
B. 3
C. 0
D. 1

Answer

To solve the problem, we need to evaluate the expression \( 2T_6 - 3T_4 + 1 \) where \( T_n = \cos^n \theta + \sin^n \theta \).

### Step-by-step Solution:

1. **Define \( T_6 \) and \( T_4 \)**:
\[ T_6 = \cos^6 \theta + \sin^6 \theta \]
\[ T_4 = \cos^4 \theta + \sin^4 \theta \]

2. **Substitute \( T_6 \) and \( T_4 \) into the expression**:
\[ 2T_6 - 3T_4 + 1 = 2(\cos^6 \theta + \sin^6 \theta) - 3(\cos^4 \theta + \sin^4 \theta) + 1 \]

3. **Use the identity for \( \cos^6 \theta + \sin^6 \theta \)**:
We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Let \( a = \cos^2 \theta \) and \( b = \sin^2 \theta \):
\[ \cos^6 \theta + \sin^6 \theta = (\cos^2 \theta + \sin^2 \theta)(\cos^4 \theta - \cos^2 \theta \sin^2 \theta + \sin^4 \theta) \]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ \cos^6 \theta + \sin^6 \theta = \cos^4 \theta - \cos^2 \theta \sin^2 \theta + \sin^4 \theta \]

4. **Substituting back into the expression**:
\[ 2T_6 = 2(\cos^4 \theta - \cos^2 \theta \sin^2 \theta + \sin^4 \theta) \]
\[ = 2\cos^4 \theta - 2\cos^2 \theta \sin^2 \theta + 2\sin^4 \theta \]

5. **Now substitute \( T_4 \)**:
\[ T_4 = \cos^4 \theta + \sin^4 \theta \]
Therefore,
\[ 3T_4 = 3(\cos^4 \theta + \sin^4 \theta) = 3\cos^4 \theta + 3\sin^4 \theta \]

6. **Combine all terms**:
\[ 2T_6 - 3T_4 + 1 = (2\cos^4 \theta - 2\cos^2 \theta \sin^2 \theta + 2\sin^4 \theta) - (3\cos^4 \theta + 3\sin^4 \theta) + 1 \]
\[ = (2\cos^4 \theta - 3\cos^4 \theta) + (2\sin^4 \theta - 3\sin^4 \theta) - 2\cos^2 \theta \sin^2 \theta + 1 \]
\[ = -\cos^4 \theta - \sin^4 \theta - 2\cos^2 \theta \sin^2 \theta + 1 \]

7. **Using the identity \( \cos^4 \theta + \sin^4 \theta = (\cos^2 \theta + \sin^2 \theta)^2 - 2\cos^2 \theta \sin^2 \theta \)**:
\[ = 1 - 2\cos^2 \theta \sin^2 \theta \]
Thus,
\[ -\cos^4 \theta - \sin^4 \theta = - (1 - 2\cos^2 \theta \sin^2 \theta) \]
Therefore,
\[ = -1 + 2\cos^2 \theta \sin^2 \theta - 2\cos^2 \theta \sin^2 \theta + 1 = 0 \]

8. **Final Result**:
\[ 2T_6 - 3T_4 + 1 = 0 \]